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These Tables codify a rudimentary level of interpretive practice for the agents ''A'' and ''B'', and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain.  Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form ‹''o'',&nbsp;''s'',&nbsp;''i''› that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

These Tables codify a rudimentary level of interpretive practice for the agents <math>\text{A}</math> and <math>\text{B}</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain.  Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)</math> that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs.  In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs.  In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''.  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''.  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

The dyadic relation that constitutes the ''denotative component'' of a sign relation ''L'' is denoted ''Den''(''L'').  Information about the denotative component of semantics can be derived from ''L'' by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, ''Proj''_''OS''&nbsp;''L'', ''L''_''OS''&nbsp;, or ''L''₁₂&nbsp;, and defined as follows:

The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L</math> is written <math>\operatorname{Den}(L)</math>.  Information about the denotative component of semantics can be derived from <math>L</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L</math>, <math>L_{OS}</math>, or <math>L_{12}</math>, and defined as follows:

: ''Den''(''L'') = ''Proj''_''OS''&nbsp;''L'' = ''L''_''OS'' = {‹''o'',&nbsp;''s''› &isin; ''O'' &times; ''S'' : ‹''o'',&nbsp;''s'',&nbsp;''i''› &isin; ''L'' for some ''i'' &isin; ''I''}.

: <math>\operatorname{Den}(L) = \operatorname{proj}_{OS} L = L_{OS} = \{ (o, s) \in O \times S : (o, s, i) \in L ~\text{for some}~ i \in I \}</math>.

Looking to the denotative aspects of the present example, various rows of the Tables specify that ''A'' uses "i" to denote ''A'' and "u" to denote ''B'', whereas ''B'' uses "i" to denote ''B'' and "u" to denote ''A''.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

Looking to the denotative aspects of the present example, various rows of the Tables specify that ''A'' uses "i" to denote ''A'' and "u" to denote ''B'', whereas ''B'' uses "i" to denote ''B'' and "u" to denote ''A''.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.